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A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cghs consisting of two disjoint triples. These were studied at length by Bra{ss} (2004) and by Aronov, Dujmovic, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.
In a generalized Turan problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path in p
We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from ${1, 2, ldots, k}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minim
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particula